In the setting with noise, the same algorithm is applied to the estimated Markov matrix \(\hat{G}\)  (see Appendix  for a simple method for obtaining \(\hat{G}\)  from \(\mathcal {D}^N_T\) ), instead of the true matrix \(G\) . In this case the Ho-Kalman Algorithm will produce estimates \(\hat{A},\hat{B},\hat{C}\)  and \(\hat{D}\) . The explicit algorithm is shown as Alg REF  (deterministic). It was shown in [1]} that the robustness of the Ho-kalman Algorithm implies the estimation error for \(A,B,C\)  and \(D\)  is bounded by \(O(\frac{1}{N^{1/4}})\) , where \(N\)  is the sample size of trajectories, i.e.,
\(\begin{aligned}\max &\left\lbrace \Vert \hat{A}-S^{-1} A S\Vert , \Vert \hat{B}-S^{-1} B\Vert ,\Vert \hat{C}-C S\Vert \right\rbrace \le \hat{C}\sqrt{\Vert G-\hat{G}\Vert } = O(\frac{1}{N^{1/4}})\end{aligned}\) 
